Volume-constrained Minimization Research Articles

Global and local energy minimizers for a nanowire growth model

We consider a model for vapor–liquid–solid growth of nanowires proposed in the physical literature. Liquid drops are described as local or global volume-constrained minimizers of the capillarity energy outside a semi-infinite convex obstacle modeling the nanowire. We first address the existence of global minimizers and then, in the case of rotationally symmetric nanowires, we investigate how the presence of a sharp edge affects the shape of local minimizers and the validity of Young’s law.

Area Quasi-minimizing Partitions with a Graphical Constraint: Relaxation and Two-Dimensional Partial Regularity

We consider a variational model for periodic partitions of the upper half-space into three regions, where two of them have prescribed volume and are subject to the geometric constraint that their union is the subgraph of a function, whose graph is a free surface. The energy of a configuration is given by the weighted sum of the areas of the interfaces between the different regions and a general volume-order term. We establish existence of minimizing configurations via relaxation of the energy involved, in any dimension. Moreover, we prove partial regularity results for volume-constrained minimizers in two space dimensions. Thin films of diblock copolymers are a possible application and motivation for considering this problem.

Conducting Flat Drops in a Confining Potential

We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and regularity estimates for volume-constrained minimizers. We also derive the Euler–Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal \(\frac{1}{2}\)-derivative of the capacitary potential.

Large mass minimizers for isoperimetric problems with integrable nonlocal potentials

This paper is concerned with volume-constrained minimization problems derived from Gamow’s liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider a large class of potentials, given by general radial nonnegative kernels which are integrable on Rn, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Contrary to the small mass case, where the nonlocal term becomes negligible compared to the perimeter, here the nonlocal term explodes compared to it. However, using the integrability of those kernels, we rewrite the problem as the minimization of the difference between the classical perimeter and a nonlocal perimeter, which converges to a multiple of the classical perimeter as the mass goes to infinity. Renormalizing to a fixed volume, we show that, if the first moment of the kernels is smaller than an explicit threshold, the problem admits minimizers of arbitrarily large mass, which contrasts with the usual case of Riesz potentials. In addition, we prove that, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, we study the stability of the ball, and show that our threshold on the first moment of the kernels is sharp in the sense that large balls go from stable to unstable. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.

Symmetry of constrained minimizers of the Cahn–Hilliard energy on the torus

We establish sufficient conditions for a function on the torus to be equal to its Steiner symmetrization and apply the result to so-called volume-constrained minimizers of the Cahn–Hilliard energy. The resulting connectedness of superlevel sets is used in two dimensions together with the Bonnesen inequality to quantitatively estimate the sphericity of minimizers. We also show how two-point rearrangements can be used to give an alternate proof of symmetry for the constrained minimizers of the Cahn–Hilliard model.

Analytical validation of a $2+1$ dimensional continuum model for epitaxial growth with elastic substrate

An analiyical validation is obtained for the evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h], introduced in [18] by W.T. Tekalign and B.J. Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, h denotes the surface height of the film, F is the Fourier transform, and a, E, r are positive material constants. Existence, uniqueness, and Lipschitz regularity in time for weak solutions are proved, under suitable assumptions on the initial datum. Introduction The epitaxial deposition of a thin film on a relatively thick substrate has gained much interest in the recent years due to its applications to semiconductor electronics and quantum dots. Roughly speaking, the morphology of the film is known to be the result of a competition between the elastic energy associated to the mismatch between film and substrate, and the surface mass transport due to the film deposition. An extensive mathematical analysis of the mechanism associated to epitaxial film growth has been carried out in [2, 3, 10, 11, 15, 16] in the context of plane linear elasticity, and regularity results have been established for volume-constrained minimizers. Short time existence for a surface diffusion type geometric evolution equation keeping into account elasticity has first been analyzed in [12] in a two-dimensional setting (see also [17]). The previous result has been recently extended in [13] to the three-dimensional case. The central aim of this work is to study existence and Lipschitz regularity in time of weak solutions to the 2+1 dimensional evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h] (0.1) derived by W.T. Tekalign and B.J. Spencer in [18], where h denotes the surface height of the film and F is the Fourier transform. The quantities a, E, r are positive material constants. To be precise, a (resp. r) is the wavenumber (resp. wetting coefficient) associated to the equation, and E is defined as E := 2μ F (1 + ν )(1− ν) (1− νF )μS , where μ and ν (resp. μ and ν) are the elastic shear modulus and the Poisson’s 2010 Mathematics Subject Classification. 35K55, 35K67, 44A15, 74K35.

Large Volume Minimizers of a Nonlocal Isoperimetric Problem: Theoretical and Numerical Approaches

We consider the volume-constrained minimization of the sum of the perimeter and the Riesz potential. We add an external potential of the form $\|{x}\|^\beta$ that provides the existence of a minimizer for any volume constraint, and we study the geometry of large volume minimizers. Then we provide a numerical method to address this variational problem.

Volume constrained minimizers of the fractional perimeter with a potential energy

We consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume integral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate.

Nonlocal quantitative isoperimetric inequalities

We show a quantitative-type isoperimetric inequality for fractional perimeters where the deficit of the $$t$$ -perimeter, up to multiplicative constants, controls from above that of the $$s$$ -perimeter, with $$s$$ smaller than $$t$$ . To do this we consider a problem of independent interest: we characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $$t$$ -perimeter and the $$s$$ -perimeter. In particular, we show that balls are the unique minimizers if the volume is sufficiently small, depending on $$t-s$$ , while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all $$s,\,t$$ . When $$s=0$$ this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.

Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies

We obtain a sharp quantitative isoperimetric inequality for nonlocal $s$-perimeters, uniform with respect to $s$ bounded away from $0$. This allows us to address local and global minimality properties of balls with respect to the volume-constrained minimization of a free energy consisting of a nonlocal $s$-perimeter plus a non-local repulsive interaction term. In the particular case $s =1$ the $s$-perimeter coincides with the classical perimeter, and our results improve the ones of Kn\"upfer and Muratov concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts.

Regularity of Free Boundaries in Anisotropic Capillarity Problems and the Validity of Young’s Law

Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young’s law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of finite perimeter (and thus to codimension one integer rectifiable currents) arising as minimizers in other variational problems with free boundaries.

A stress–based approach to the optimal design of structures with unilateral behavior of material or supports

The paper presents a stress-based approach that copes with the optimal design of truss-like elastic structures in case of unilateral behavior of material or ground supports. The conventional volume-constrained minimization of compliance is coupled with a set of local stress constraints that are enforced, all over the domain or along prescribed boundaries, to control the arising of members with tension-only or compression-only strength. A Drucker–Prager failure criterion is formulated to provide a smooth approximation of the no-tension or no-compression conditions governing the stress field. A selection strategy is implemented to handle efficiently the multi-constrained formulation that is solved through mathematical programming. Benchmark examples are investigated to discuss the features of the achieved optimal designs, as compared with problems involving material and ground supports with equal behavior in tension and compression. Numerical simulations show that a limited set of constraints is needed in the first iterations to steer the solution of the energy-driven optimization towards designs accounting for the prescribed assumption of unilateral strength.

Volume-constrained minimizers for the prescribed curvature problem in periodic media

We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a suitable Wulff Shape, when the volume tends to infinity.